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Kinetic & Related Models

Open Access Articles

A deterministic-stochastic method for computing the Boltzmann collision integral in $\mathcal{O}(MN)$ operations
Alexander Alekseenko, Truong Nguyen and Aihua Wood
2018, 11(5): 1211-1234 doi: 10.3934/krm.2018047 +[Abstract](4913) +[HTML](1279) +[PDF](2466.17KB)

We developed and implemented a numerical algorithm for evaluating the Boltzmann collision integral with \begin{document}$O(MN)$\end{document} operations, where \begin{document}$N$\end{document} is the number of the discrete velocity points and \begin{document}$M <N$\end{document}. At the base of the algorithm are nodal-discontinuous Galerkin discretizations of the collision operator on uniform grids and a bilinear convolution form of the Galerkin projection of the collision operator. Efficiency of the algorithm is achieved by applying singular value decomposition compression of the discrete collision kernel and by approximating the kinetic solution by a sum of Maxwellian streams using a stochastic likelihood maximization algorithm. Accuracy of the method is established on solutions to the problem of spatially homogeneous relaxation.

Mathematical modeling of a discontinuous solution of the generalized Poisson-Nernst-Planck problem in a two-phase medium
Victor A. Kovtunenko and Anna V. Zubkova
2018, 11(1): 119-135 doi: 10.3934/krm.2018007 +[Abstract](5003) +[HTML](1045) +[PDF](452.3KB)

In this paper a mathematical model generalizing Poisson-Nernst-Planck system is considered. The generalized model presents electrokinetics of species in a two-phase medium consisted of solid particles and a pore space. The governing relations describe cross-diffusion of the charged species together with the overall electrostatic potential. At the interface between the pore and the solid phases nonlinear electro-chemical reactions are taken into account provided by jumps of field variables. The main advantage of the generalized model is that the total mass balance is kept within our setting. As the result of the variational approach, well-posedness properties of a discontinuous solution of the problem are demonstrated and supported by the energy and entropy estimates.

Numerical study of a particle method for gradient flows
José Antonio Carrillo, Yanghong Huang, Francesco Saverio Patacchini and Gershon Wolansky
2017, 10(3): 613-641 doi: 10.3934/krm.2017025 +[Abstract](3479) +[HTML](741) +[PDF](728.3KB)

We study the numerical behaviour of a particle method for gradient flows involving linear and nonlinear diffusion. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles. The resulting scheme preserves the gradient flow structure at the particle level and enables us to obtain a gradient descent formulation after time discretisation. We give several simulations to illustrate the validity of this method, as well as a detailed study of one-dimensional aggregation-diffusion equations.

Self-organized hydrodynamics with density-dependent velocity
Pierre Degond, Silke Henkes and Hui Yu
2017, 10(1): 193-213 doi: 10.3934/krm.2017008 +[Abstract](3204) +[HTML](1125) +[PDF](651.5KB)

Motivated by recent experimental and computational results that show a motility-induced clustering transition in self-propelled particle systems, we study an individual model and its corresponding Self-Organized Hydrodynamic model for collective behaviour that incorporates a density-dependent velocity, as well as inter-particle alignment. The modal analysis of the hydrodynamic model elucidates the relationship between the stability of the equilibria and the changing velocity, and the formation of clusters. We find, in agreement with earlier results for non-aligning particles, that the key criterion for stability is \begin{document} $(ρ v(ρ))'≥q 0$ \end{document}, i.e. a nondecreasing mass flux \begin{document} $ρ v(ρ)$ \end{document} with respect to the density. Numerical simulation for both the individual and hydrodynamic models with a velocity function inspired by experiment demonstrates the validity of the theoretical results.

Anton Arnold, Paola Pietra and Giuseppe Toscani
2017, 10(1): i-ii doi: 10.3934/krm.201701i +[Abstract](1618) +[HTML](1228) +[PDF](1249.2KB)
Kazuo Aoki, Pierre Degond and Tong Yang
2013, 6(4): i-ii doi: 10.3934/krm.2013.6.4i +[Abstract](1460) +[PDF](504.2KB)
In 2012, we are saddened by the loss of our friend and a great mathematician, Seiji Ukai, who passed away in November. Seiji was a leading expert in the area of kinetic theory and fluid mechanics, and was always kind and humble so that he was beloved by his colleagues, collaborators and researchers in these research areas. As Cédric Villani said in his article, ``Good-bye my friend...(In Memoriam Seiji Ukai)", ``He was a great mathematician and a beautiful soul".

For more information please click the “Full Text” above.
On the dynamics of social conflicts: Looking for the black swan
Nicola Bellomo, Miguel A. Herrero and Andrea Tosin
2013, 6(3): 459-479 doi: 10.3934/krm.2013.6.459 +[Abstract](3004) +[PDF](702.8KB)
This paper deals with the modeling of social competition, possibly resulting in the onset of extreme conflicts. More precisely, we discuss models describing the interplay between individual competition for wealth distribution that, when coupled with political stances coming from support or opposition to a Government, may give rise to strongly self-enhanced effects. The latter may be thought of as the early stages of massive unpredictable events known as Black Swans, although no analysis of any fully-developed Black Swan is provided here. Our approach makes use of the framework of the kinetic theory for active particles, where nonlinear interactions among subjects are modeled according to game-theoretical principles.
Paola Pietra, Eric Polizzi, Fabrice Deluzet, Jihène Kéfi, Olivier Pinaud, Claudia Negulescu, Nicolas Vauchelet, Raymond El Hajj, Clément Jourdana and Stefan Possanner
2011, 4(4): i-iii doi: 10.3934/krm.2011.4.4i +[Abstract](1397) +[PDF](892.5KB)
On July 5-th 2010, Naoufel Ben Abdallah tragically passed away at the age of 41. He was an extremely talented mathematician with a deep interest in applications, and an uncommon ability in creating links for interdisciplinary research.

For more information please click the "Full Text" above.
Maria Lampis and Mario Pulvirenti
2011, 4(1): i-v doi: 10.3934/krm.2011.4.1i +[Abstract](1574) +[PDF](2567.3KB)
On January 7-th, 2010, Carlo Cercignani, Professor of Mathematical Physics at Politecnico di Milano, passed away in Milan, after a long illness.
   If one ought to indicate a single man as a reference point for the development of kinetic theory in the last 50 years from a mathematical, physical, historical and, more generally, cultural point of view, there is no doubt that everybody would think of Carlo Cercignani. We have not only lost a very reputed and loved colleague, but also a pivotal figure for all of us working in the field.

For more information please click the “Full Text” above.
José Antonio Carrillo, Peter Markowich and Lorenzo Pareschi
2010, 3(1): i-i doi: 10.3934/krm.2010.3.1i +[Abstract](1491) +[PDF](27.1KB)
The idea of putting together a special issue of KRM in honor of Giuseppe Toscani's 60$th$ birthday was received with great enthusiasm by the "kinetic" community and by Giuseppe's scientific colleagues and friends from other mathematical communities. There is a common feeling throughout that we have learned a great deal from the ideas, the methodologies and the scientific conclusions of Giuseppe Toscani's work.
   The papers related to Toscani's research in this special issue are authored by collaborators, friends and students of Giuseppe. The topics include classical and non classical applications of kinetic theory, entropy-entropy dissipation methods, asymptotic techniques and numerical simulations.The articles in this volume are ordered alphabetically by names of authors.

For more information please click the “Full Text” above.
Existence and sharp localization in velocity of small-amplitude Boltzmann shocks
Guy Métivier and K. Zumbrun
2009, 2(4): 667-705 doi: 10.3934/krm.2009.2.667 +[Abstract](1781) +[PDF](465.9KB)
Using a weighted $H^s$-contraction mapping argument based on the macro-micro decomposition of Liu and Yu, we give an elementary proof of existence, with sharp rates of decay and distance from the Chapman-Enskog approximation, of small-amplitude shock profiles of the Boltzmann equation with hard-sphere potential, recovering and slightly sharpening results obtained by Caflisch and Nicolaenko using different techniques. A key technical point in both analyses is that the linearized collision operator $L$ is negative definite on its range, not only in the standard square-root Maxwellian weighted norm for which it is self-adjoint, but also in norms with nearby weights. Exploring this issue further, we show that $L$ is negative definite on its range in a much wider class of norms including norms with weights asymptotic nearly to a full Maxwellian rather than its square root. This yields sharp localization in velocity at near-Maxwellian rate, rather than the square-root rate obtained in previous analyses.

2019  Impact Factor: 1.311




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